Decision Under Normative Uncertainty Franz Dietrich Brian Jabarian PSE & CES & CNRS U. Paris 1 & PSE Second Workshop on Coping With Heterogeneous Opinions Paris School of Economics 29 November 2018
Empirical vs. normative uncertainty • Classical empirical uncertainty: uncertainty about empirical facts. — Ex : Does a medical treatment cure the patient? What are the side e ff ects? • Normative uncertainty: uncertainty about value facts. — Ex : Is curing the patient worth the side e ff ects? How much does the patient’s will count? What is the correct inequality aversion? — More generally : What is the correct normative theory? (Is it utilitarianism, some egalitarianism, some prioritarianism, some deontology, ...?)
Should we close down nuclear plants? Two dimensions of this debate: • empirical uncertainty : Will there be earth quakes? human errors? technological progress? etc. • normative uncertainty : How evaluate burdens for future gen- erations? What is the correct intergenerational discounting factor? How trade o ff between quality of life and probability of death in accidents? etc.
Goal: incorporate normative uncertainty into decision models
Why important? Understanding both sides of (social and internal) deliberation
‘Value’ could stand for... • individual well-being, • social welfare, • moral value, • legal value, • artistic value, • ...
Conceptualizing normative uncertainty within Savage’s framework Coming from Savage’s decision theory, one might think of • empirical uncertainty as uncertainty about the nature state (interpreted as the empirical state of the world) • normative uncertainty as uncertainty about the value/utility of consequences. Classical EU-agents have only empirical uncertainty: they do not know the state, but know (‘have’) exact utilities of conse- quences.
Note our cognitive re-interpretation of ‘utility’ Utility as Utility as uncertainty introducing reinterpretation desire believed value normative uncertainty about true value Figure 1: In 2 steps in normative uncertainty
From a Humean belief/desire model to a cognitivist model
Normative uncertainty: philosophically meaningful? • Normative uncertainty presupposes (beliefs about) normative facts. • ‘Normative facts’?? Don’t worry: these facts can be objective or subjective, universal or relative, ... I’ll spare you with philosophical debates around ‘facts’.
Normative uncertainty: formally di ff erent? • A legitimate question! (Which I had too, 1 year ago.) • Modelling normative uncertainty as ordinary choice-theoretic uncertainty fails. • So: normative uncertainty di ff ers not just interpretively, but also formally.
Philosophers have started formal work on normative uncertainty • MacAskill (2014, 2016), Greaves & Ord (2018), Lockhart (2000), Ross (2006), Sepielli (2009), Barry & Tomlin (2016) • Some points of focus: — cardinal vs ordinal value — comparable vs non-comparable value — individual vs collective choice — consequentialist vs non-consequentialist evaluations
The Question • How evaluate options under normative uncertainty? — What’s the ‘meta-value’ under uncertainty about ‘1 st -order value’?
Plan 1. The classical ‘expected-value theory’ 2. An alternative ‘impartial value theory’
Options and Valuations Consider: • a set of ‘ options ’, the objects of evaluation — choice options, policy measures, social arrangements, in- come distributions, ... — (For now we set aside empirical uncertainty. But in principle options could contain empirical uncertainty.)
Valuations • a fi nite set V of ‘ valuations ’ , assigning to each option ∈ its value ( ) in R . — They might represent rival normative theories, normative intuitions, value judgments, ‘social preferences’, ... — V might consist of: ∗ a utilitarian and a Rawlsian valuation, or ∗ ‘similar’ valuations di ff ering in a parameter, e.g., in a discounting factor, or inequality-aversion degree, or pri- oritarian degree, ...
Value versus vNM utility
Beliefs about value Consider further: • a probability function assigning to each valuation in V its subjective correctness probability ( ) ≥ 0 , where P ∈ V ( ) = 1 .
Meta-theories • What is the overall value of each option, given one’s normative uncertainty? • An answer is a ‘meta-’valuation , assigning to each option in its ‘overall’ value. • Prominent proposal: the expected-value theory ‘ ’ which valuates each option ∈ by its expected value: X ( ) = ( ) ( ) ∈ V
EV is neutral to normative risk
Neutrality to normative risk is implausible if aversion to empirical risk is certainly correct
What does it mean that aversion to empirical risk is certainly correct? • Assume options in contain empirical uncertainty. say they are vNM lotteries on a set of ‘outcomes’. • The value of an outcome in is the value of the sure lottery in which yields . • The risk attititude of a valuation ∈ V is given by how ( ) P compares to the expected outcome-value ∈ ( ) ( ) . P • Risk-aversion is certainty correct if ( ) ∈ ( ) ( ) for all non-sure lotteries and all ∈ V s.t. ( ) 6 = 0 .
The attitude of EV to empirical risk is impartial : it is guided by the risk-attitudinal beliefs • is neutral (averse, prone) to empirical risk if all ∈ V of non-zero correctness probability ( ) are risk-neutral (- averse, -prone). Formally, evaluates options without nor- mative risk at (below, above) the option’s expected outcome value if each ∈ V s.t. ( ) 6 = 0 does so. • ‘Impartiality’ of risk attotides can be de fi ned precisely.
In the paper we de fi ne 3 alternatives to EV, with di ff erent risk attitudes neutral to nor. risk impartial to nor. risk neutral to emp. risk ‘fully expectational value’ ‘dual expected value’ impartial to emp. risk ‘expected value’ ‘impartial value’
Our favourite: the impartial value theory. How is it de fi ned?
Value prospects • A value prospect is a lottery over value levels in R . • Each option ∈ generates two types of value prospect, depending on whether we consider just empirical or also nor- mative uncertainty: — ’s value prospect under ∈ V is denoted and given by: ( ) = prob. of an outcome of value under X = ( ) ∈ : ( )= — ’s value prospect simpliciter is denoted and given by: ( ) = prob. of an outcome of value X = ( ) ( ) | {z } ( ) ∈ V× : ( )= prob. of ( )
Impartial Value de fi ned • Each valuation in V can be taken to evaluate not just options , but also value prospects : 1 ( ) = value ( ) of options with value prospect = • The impartial theory ‘ ’ evaluates each option ∈ by the expected evaluation of its value prospect: X ( ) = ( ) ( ) . ∈ V 1 This de fi nition presupposes a technical assumption: for each valuation in V and value prospect , let there exist a corresponding option in whose value prospect is , and moreover let any two such options in have same value ( ) .
IV versus EV • Assume that being risk-averse is certainly correct, i.e., only risk-averse theories in V have positive probability. P • The expected value ( ) = ∈ V ( ) ( ) contains a risk premium for empirical risk, because each ‘ ( ) ’ contains a premium for the ( empirical ) risk in . P • The impartial value ( ) = ∈ V ( ) ( ) contains a risk premium for empirical and normative risk, because each ‘ ( ) ’ contains a premium for the ( empirical and normative ) risk in .
Ex-ante vs. ex-post approach • Famous question in ethics and aggregation theory: should competing evaluations of uncertain prospects be aggregated before or after resolution of uncertainty? (See, e.g., Fleurbaey 2010, Fleurbaey and Zuber 2017.) • We have two types of uncertainty, so four approaches: normatively ex-post normatively ex-ante empirically ex-post fully expectational value dual expected value empirically ex-ante expected value impartial value
Why do we base IV on an expectation? • Is not risk-neutral through the back door, through taking the expectation of the ( ) ( ∈ V )? • No, because each ( ) ( ∈ V ) already contains a premium for all the risk in the option , empirical and normative. De fi n- ing ( ) as a value below that expectation would amount to a ‘double risk premium’.
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